Finite moments testing in a general class of nonlinear time series models

We investigate the problem of testing the finiteness of moments for a class of semi-parametric time series encompassing many commonly used specifications. The existence of positive-power moments of the strictly stationary solution is characterized by the Moment Determining Function (MDF) of the model, which depends on the parameter driving the dynamics and on the distribution of the innovations. We establish the asymptotic distribution of the empirical MDF, from which tests of moments are deduced. Alternative tests based on estimation of the Maximal Moment Exponent (MME) are studied. Power comparisons based on local alternatives and the Bahadur approach are proposed. We provide an illustration on real financial data and show that semi-parametric estimation of the MME provides an interesting alternative to Hill's nonparametric estimator of the tail index.